Une équation simple au comportement complexe

According to WJ Freeman experiments and simulations, arousal level can be simulated by the "complexity parameter" in his numerical model. When arousal or the complexity parameter increase, we saw that electrophysiologic responses (real or simulated) pass by simple attractors (sleep), then limit cycle (waking rest), then strange attractors (motivated state), and finally by limit cycle (seizures).

Writing a code for freeman equations is rather difficult, but another (obviously) very simple equation perfectly illustrates such a behavior:

xt+1 = r . xt (1 - xt)

in which, xt may be a potential recorded at time t, and r the complexity parameter (or arousal level).We must choose 0<x<1, and 0<r<4